The problem of distributed optimization requires a group of networked agents to compute a parameter that minimizes the average of their local cost functions. While there are a variety of distributed optimization algorithms that can solve this problem, they are typically vulnerable to ``Byzantine'' agents that do not follow the algorithm. Recent attempts to address this issue focus on single dimensional functions, or assume certain statistical properties of the functions at the agents. In this paper, we provide two resilient, scalable, distributed optimization algorithms for multi-dimensional functions. Our schemes involve two filters, (1) a distance-based filter and (2) a min-max filter, which each remove neighborhood states that are extreme (defined precisely in our algorithms) at each iteration. We show that these algorithms can mitigate the impact of up to $F$ (unknown) Byzantine agents in the neighborhood of each regular agent. In particular, we show that if the network topology satisfies certain conditions, all of the regular agents' states are guaranteed to converge to a bounded region that contains the minimizer of the average of the regular agents' functions.

翻译：分布式优化问题要求一组联网智能体计算一个参数，以最小化其局部成本函数的平均值。尽管存在多种可解决该问题的分布式优化算法，但通常易受不遵循算法的“拜占庭”智能体攻击。近期应对该问题的尝试聚焦于单维函数，或假设智能体处函数具有特定统计特性。本文针对多维函数提出两种具有鲁棒性、可扩展的分布式优化算法。我们的方案包含两种滤波器：（1）基于距离的滤波器和（2）最小-最大滤波器，二者在每次迭代中移除邻域内的极端状态（具体定义见算法）。我们证明，这些算法可将每个正常智能体邻域内至多$F$个未知拜占庭智能体的影响降至最低。特别地，我们表明当网络拓扑满足特定条件时，所有正常智能体的状态可保证收敛至包含正常智能体函数平均值最小化器的有界区域。